Z couplings to pseudo-Goldstone bosons within extended technicolor

Nuclear Physics B340 (1990) 1—32 North-Holland

Z COUPLINGS TO PSEUDO-GOLDSTONE BOSONS WITHIN EXTENDED TECHNICOLOR M. SOLDATE and Raman SUNDRUM Centerfor Theoretical Physics, Yale University, 217 Prospect Street, New Haven, CT 06511, USA Received 19 December 1989

The couplings of the Z to pseudo-Goldstone particles occurring within technicolor theories are examined using the chiral lagrangian technique. Particular attention is paid to the effects of the four-technifermion interactions of extended technicolor. Interactions of this type can give the pseudo-Goldstone bosons their masses and also can induce couplings of the form ZCZCPO, Z~(P 0~9”P~ — P~9’~P0) and Z~’W~P. We show however that none of the four-Fermi interactions which induce the neutral pseudo-Goldstone ZCZ~’POcouplings contribute to the average mass squared of the pseudo-Goldstone bosons to first order, so that these couplings can be appreciable even for light pseudo-Goldstone bosons. The Z,~Z~P0couplings are Higgs-like but for symmetry reasons cannot occur in pure technicolor models without four-technifermion interactions. The Z,~(PO(9eP6— P~~~’P~) couplings are also Higgs-like, and unless unusual hypercharge assignments are given to the technifermions, they too are forbidden in the absence of four-technifermion interactions. When these four-Fermi couplings are taken into account, the above vertices can occur, and there exists the possibility that they can approach strengths comparable to their standard Higgs counterparts in models where the number of technifermion flavors is reasonably large. This can make the experimental differentiation of pseudo-Goldstone particles from Higgs particles more difficult. The vertex Z”W~Pis notably not Higgs-like and therefore we have found it relevant also to study this coupling.

1. Introduction There is wide (but not universal) agreement that the W and Z acquire mass through the Higgs mechanism. A central aspect of analyzing electroweak symmetry breaking (EWSB) is then to understand the forces which give rise to the three required Goldstone bosons. In many descriptions of EWSB, there are accompanying spin-U particles, whose observation could well provide significant insights into the mechanism of EWSB. The standard Higgs boson [1] provides a familiar example of one sort of such particle; qualitatively, it is a “radial” field oscillation. The usual supersymmetric extensions of the standard model have a number of particles of this type [2]. A second sort corresponds qualitatively to oscillations in approximately flat directions of a potential; such a particle is a pseudo-Goldstone 0550-3213/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

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boson (PGB) [3,4]. Technicolor theories [5] contain such particles if there is more than one SU(2)L-doublet of technifermions. (Models in which the masses of the W and Z are not due to the Higgs mechanism also can have interesting spin-U particles in their spectra, see e.g. ref. [61.) A basic phenomenological task is to distinguish between Higgs particles and PGBs (of technicolor theories). Their couplings can differ markedly. A general remark of particular relevance is that Goldstone bosons have derivative couplings [71while Higgs particles do not. However, the explicit symmetry breaking which gives mass to a pseudo-Goldstone boson can induce nonderivative couplings as well. Therefore, one expects that generically the phenomenological discrimination of a Higgs particle from a PGB of the same mass will become more difficult as the mass increases. This issue is the primary general concern illustrated here. However, we will be examining a situation in which this expected link between the mass of a PGB and the strength of its nonderivative couplings may fail due to additional symmetry considerations. To first order, we will find interactions which induce nonderivative couplings for a PGB but which do not necessarily increase the PGB’s mass as the coupling strength is increased. The experimental search for spin-U particles associated with EWSB will rely heavily on the couplings of these particles to gauge bosons. Couplings to the Z will be especially important in the near future. The first extensive searches will be at the Z factories, LEP and SLC. For example, with expected luminosities it should be possible to search for a particle with the couplings of the standard Higgs up to a mass of approximately 40 GeV. The dominant vertex [81for its production is of the form H0Z~Z~, a nonderivative coupling. LEP 200 will exploit the same vertex to search for the standard Higgs with mass up to approximately 80 GeV [9]. Other interesting searches at these e~e colliders are for two oppositely1H— charged scalars H3~H~) or or two distinctH(~9’~H neutral scalars from couplings such as Z1~(H~0’ Z~,(HO0’~Hd 2 0). At higher energies the couplings H0W~ W~’and HØZ~Z’ are exploited in the WW/ZZ fusion production mechanism for a heavy standardmodel Higgs at both e~e and hadron colliders [10]. These are some of the more prominent examples of production via couplings of gauge particles and fundamental scalars. The coupling Z’~W~H~ is notably absent in multidoublet models [11, 121. It is relevant phenomenologically to understand the extent to which analogous couplings of gauge particles and PGBs can arise. The theoretical descriptions of Higgs particles and the PGBs of technicolor are rather different. The sector responsible for EWSB should exhibit presumably at least an (approximate) SU(2)L x SU(2) global symmetry spontaneously broken to the custodial SU(2) subgroup in order to give p 1 as a natural relation [131.The realization of the symmetries involving Goldstone boson fields is either linear or nonlinear, the simplest version [SU(2)L X SU(2)R —s SU(2)~]being the linear [14] or nonlinear [15, 161 a- model familiar from pion dynamics. A basic consideration is that a nonlinear a- model is nonrenormalizable, at least perturbatively. Fundamen—

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tal Goldstone bosons and other scalars are described by the familiar linear amodel or elaborations of it, due to the requirement of renormalizability of the theory. The standard Higgs boson corresponds of course to the a- field of the original linear a- model. The Goldstone boson fields of the nonlinear a- model correspond to composite particles. (Composite Goldstone bosons may be described also in a linear realization at sufficiently low scales; see ultracolor models [171.) Technicolor is a natural candidate for the renormalizable dynamics underlying the description of EWSB using a nonlinear a- model. In technicolor, the SU(2)L x SU(2) global symmetry is interpreted as the SU(2)L X SU(2)R chiral symmetry of one SU(2)L doublet and two right-handed singlets of technifermions. Extensions of this chiral symmetry can be encorporated into the nonlinear a- model, thereby describing the dynamics of additional Goldstone bosons. Phenomenologically, these probably should acquire mass from some source of explicit symmetry breaking to become PGBs. The masses of spin-U particles associated with EWSB and their interactions with gauge particles have been studied extensively. A few references for these issues in multi-doublet Higgs models are refs. [18, 191, the latter pertaining to the more restricted situation appropriate for supersymmetry breaking near the weak scale. The properties of PGBs in technicolor models including the gauge interactions of the standard model have also received careful attention. Refs. [4, 20] discuss the masses that technicolor PGBs acquire through standard-model gauge interactions. The first discussions of PGB interactions in the context of technicolor are [21, 22]. In ref. [23] the absence of the coupling ZILZ’~P()was emphasized and the principal phenomenological implications were discussed, along with other related matters. The couplings of standard-model gauge bosons to PGBs which are not anomalyinduced were analyzed in remarkable generality in ref. [24], and applied to PGB production in Z° decays. In particular, it was observed that the coupling ZM(PJ~P+ P~a~P_) appears with the standard strength of charged fundamental scalars, while other couplings such as Z~Z’~Pa0nd Z~L(POä’2P(~ P(çd”P0), whose —

Higgs-scalar analogs occur typically, are absent to lowest order in the derivative expansion, in general for the first coupling and in the one-generation model for the second. Analysis of anomaly-induced couplings appears in ref. [25]; these effects are typically small in part because of one-loop suppression factors. One of the objects of this paper is to provide an extensive analysis of the symmetries which can restrict phenomenologically relevant PGB-gauge boson couplings arising from technicolor and standard-model gauge interactions alone. In order to give ordinary fermions (quarks and leptons) mass in the context of technicolor, it is necessary to introduce into the technicolor lagrangian effective four-Fermi operators coupling technifermions to ordinary fermions. Generically, it is also expected that in addition there will be four-Fermi couplings of ordinary fermions alone, and technifermions alone. The first sort is responsible for the flavor-changing neutral currents which have plagued the technicolor approach

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[26, 27]. The dynamics underlying the effective four-Fermi operators has been suggested to be either massive gauge-particle exchange, as in extended technicolor (ETC) [26, 28], or residual interactions due to the hypothesized compositeness of ordinary fermions and technifermions [29]. In this paper we will discuss the effects of four-technifermion effective operators on the masses of PGBs [26,30] and, more novelly, on the interactions of gauge bosons and PGBs. The requirement of the presence of certain four-technifermion operators to give mass to specific PGBs of technicolor was recognized early in the development of ETC [26]. The possibility that ETC interactions could influence strongly the masses of PGBs was proposed by Appelquist and Wijewardhana [31] in the context of “walking” technicolor [31, 32]. They found that the lightest color-singlet PGBs could be rather heavy ( 50 6eV). Our work is motivated by their results; by extension, it is possible that four-technifermion operators can influence strongly the interactions of gauge particles and PGBs. However, as our analysis is based on a phenomenological lagrangian approach, low-energy consequences of symmetry considerations can be found without explicit reference to subtleties in the underlying technicolor dynamics. Consequently, the analysis is of more general applicability. This paper is concerned particularly with the possibility that four-technifermion operators can induce appreciable Higgs-like couplings of PGBs and electroweak gauge bosons, beyond the coupling of charged PGB pairs to the Z. In the context of extended technicolor theories with SU(NTC) as the technicolor gauge group, we use a phenomenological lagrangian to give general prescriptions for inducing the Higgs-like couplings Z,~ZI~P() and Z,J(P0a’~P( P~’3~P0), as well as Z’~W~P~. The —

strengths of these couplings are determined by parameters depending on the details of technicolor dynamics and the strengths of four-technifermion effective operators, and hence are model dependent. An immediate concern is whether or not a neutral PGB light enough to be produced in Z decays can couple appreciably to the Z, in particular as Z~Z’~P0. This coupling does not occur in leading order through technicolor and electroweak interactions [21,24], and the anomaly-induced coupling is very weak. Also, the estimate for the 2v, ETC-induced which appears in the literature [11] where ATC is acoupling technicolor scale. This has led to the belief thatis O(m~,/A?~~)g the “Higgs-like” signal of a light neutral scalar in Z decays would rule out technicolor. However, we will argue generally from symmetry considerations that the strength of the ETC-induced coupling is not fixed by the mass of the PGB produced, and will describe circumstances under which the strength of the coupling for a relatively light PGB might approach standard Higgs strength. The non-Higgs-like couplings Z’~WP~also would be significant under these circumstances. Thus, because the couplings of the neutral PGBs might mimic those of Higgs particles, the Z~W~P~ couplings may provide a less ambiguous phenomenological signature for technicolor theories.

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The phenomenological operators containing the interesting vertices have coupling constants which are not determined by the phenomenological lagrangian approach. Thus the absolute strengths of the vertices cannot be found. However, the operators containing the vertices are an order higher in the derivative expansion of the phenomenological lagrangian than the operators which contain the ETC-induced mass terms for the PGBs, so the two sets of operators can be expected to have coupling constants roughly related by the square of the chiral symmetry breaking scale, the scale which governs the convergence of the derivative expansion. It is this fact that allows us to obtain reasonable guesses for the strength of the vertices in terms of the typical ETC-induced mass of the PGBs. In order to do this we first estimate the size of the chiral symmetry breaking scale, using standard techniques adapted for our purposes. We find that in models with many technifermion flavors and typical ETC-induced PGB masses in the hundreds of GeVs, the strengths of the vertices we are interested in can approach Higgs-like values. However, the link between the coupling strength of a vertex and the masses of the particular PGBs in it is apparently not tight and it seems possible therefore to have light (say 30 GeV) PGBs with Higgs-like couplings in models where there are many PGBs with significant variations in their masses so that the average mass squared is large. The present analysis will be carried out in the context of ETC, for simplicity. ETC four-technifermion interactions are of the current—current form alone. A more general Lorentz structure can arise in composite models. Our analysis can be extended to this case; the complications appear to be purely technical. The paper is organized as follows. In sect. 2 we outline the construction of the phenomenological lagrangian in the absence of the four-Fermi perturbations. In sect. 3 the four-technifermion interactions are incorporated in the phenomenological description and their effects are parametrized. In sect. 4 the chiral symmetry breaking scale which determines the goodness of the derivative expansion of the phenomenological theory is discussed when there are many flavors of technifermions in the fundamental theory. It will turn out to be important in fixing the strengths of the PGB couplings to the Z. In sect. 5 the PGBs are parametrized and their quantum numbers and symmetry properties discussed. In sect. 6 we introduce the vertices coupling the Z to the PGBs which are the most relevant phenomenologically. Their amplitudes in the phenomenological lagrangian are computed in sect. 7. Sects. 8 and 9 deal with estimating the relationships possible between the masses and couplings to the Z of the PGBs. In sect. 10 the effects of the four-technifermion interactions on the weak gauge boson masses and the p relation are discussed. Sect. 11 deals with a certain peculiar subclass of PGBs ignored till then. In sect. 12 the effects of QCD on the theory are explained. In sect. 13 we present the possible strengths for the interesting vertices and compare them with their Higgs counterparts. Various technical matters are discussed in the appendices.

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2. The phenomenological lagrangian The symmetries and low-energy dynamics of technicolor PGBs and the electroweak gauge bosons can be described by means of a phenomenological Iagrangian. For simplicity we will assume that the underlying technicolor theory is a vector-like gauge theory with gauge group SU(NTC). The technifermions are assumed to transform as ND (left-handed) doublets and 2ND (right-handed) singlets under SU(2)L, with hypercharge assignments such that the technifermion condensates leave U(1)EM unbroken. The phenomenological lagrangian is constructed from the electroweak gauge fields and an SU(2ND)-valued field U(x), representing the Goldstone modes arising from the spontaneous breaking of the chiral symmetry SU(2ND)L X SU(2ND)R to SU(2ND)v in the technicolor theory. We first describe the allowed terms in the phenomenological lagrangian, —~h’ when ETC interactions are excluded from the fundamental lagrangian, /. The effects of ETC interactions will be included as a perturbation to / and correspondingly to in the next section. The fundamental lagrangian is thus taken to be ~~/=

(2.1)

4LJ~J)+flhJyD/~J,

where the are the field strengths for SU(NTC) x SU(2)~x U(1)~. Color interactions are an inessential complication for our analysis, and will be reinstated 2ND)L X SU(2ND)R as U(x) LU(x)Rt. easily later. transforms under SU( Because the U(x) fundamental lagrangian is SU(2ND)L X SU(2N~)~-symmetric when the electroweak gauge couplings are set to zero, in this limit the operators occurring in are required to be invariant under these chiral transformations. Further, —~h is required to be electroweak gauge invariant and CF invariant, reflecting these two aspects of the fundamental lagrangian. The explicit form of the parametrization of U(x) in terms of pseudoscalar fields and their transformations under CP will be described in sect. 5. Of the operators permitted in those with the lowest dimension are expected to be the most significant for low-energy dynamics. Thus we can write —*

=

f2(Tr[(D

U)(D~Ut)] +

. ..

}.

(2.2)

The action of the covariant derivative is given by D~U ÔU + iZ~(L~U — =

UR~)

—

iW~Lw-U iW~L~+U L4M[Q, —

—

U],

(2.3)

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where the electroweak generators (2ND X 2N~matrices) are (LZ)am,~j3m 0

=

~gcos

OWTmaj~

—

~

(Rz)amj~m~

~g’sin —

(Lw—)am~,pm~= ~

=

OWT,~mCY~,

~g’sin6wr,~~Y~,

~-~r-g(r1

—

1T2)mmj~afi,

(2.4)

+ T~m~Yap).

Qam,~,i3rn~ =

Here a and f3 run from 1 to ND specifying a “doublet”, while m~and m4 run from 1 to 2 specifying a member of a “doublet”; for this purpose right-handed (as well as left-handed) technifermions are grouped into pairs. The notation conveniently encorporates the nature of the electroweak interactions. Y is a diagonal ND X ND matrix, with 1’~(no sum over a) the hypercharge of the ath 2)L left-handed and U(1)~ doublet. The couplings g and g’ are the gauge couplings of SU( respectively, in the conventions of ref. [33]. The coefficient f is normally fixed by requiring that the leading operator of eq. (2.2) be responsible for giving the W and Z bosons their standard masses through the Higgs mechanism. This yields the relation f2=v2/ND,

v~246GeV.

(2.5)

A significant set of interactions not appearing explicitly in eq. (2.2) is the gauged Wess—Zumino term [34]. It describes in particular the leading interactions of PGBs and gauge bosons induced by anomaly effects [25]. As discussed above, the interactions of PGBs and standard-model gauge bosons arising from the first term of eq. (2.2) and the gauged Wess—Zumino term have been studied extensively in the literature. We will comment on them in sect. 6. 3. The effects of four-Fermi interactions The fundamental lagrangian will include in general four-Fermi couplings between technifermions induced by ETC interactions. The four-Fermi operators are of course SU(NTC) and SU(2)L x U(1)~. invariant. Vacuum alignment will be discussed in sect. 8. We will work to first order in these couplings; there are no obvious inconsistencies with this approximation. Some four-technifermion operators which we include in the perturbation, &/‘, to the fundamental lagrangian are ~r(1) ï~ ABCD LAB

,~(t)

n~

t

j’(I)

j’~~ i

Rpc 0’

ABCD

RAn

R~co’

~ABCD

LAB Lp~D’

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where ~I’LAY ~‘L~ and ~RAY~RB are technicolor-singlet currents. Here, A,. .., D run from 1,.. . , 2ND; the expanded notation of sect. 2 is unnecessary for the moment. The list of the perturbing four-technifermions operators is completed by including analogous operators which are SU(N~~)-singlet combinations formed from the product of two currents transforming as adjoint representations under SU(NTC), e.g. ~ where AB are the generators of SU(NTC). The requirements of SU(2)L>< U(1)~invariance impose certain constraints on the coefficients ~ci~1, ~ and ~ j 1,2. The transformation properties of the operators under SU(2ND)L X SU(2ND)R, e.g. =

=

=

J’~J LAB

P.CD

—*L BE LtF/i R

Rt

~ LFE J R~zH(;),

(3.1)

will be used to constrain the corresponding perturbations in the phenomenological lagrangian, 8—~h When SU(2)L x U(1)~ couplings are turned off the phenomenological lagrangian is perturbed to first order by a linear combination of operators transforming under SU(2ND)L x SU(2ND)R in the same way as four-Fermi terms appearing in the fundamental lagrangian. In these operators ordinary derivatives should be replaced by covariant derivatives when electroweak gauge couplings are turned on. In general, we will assume that no electroweak gauge bosons are integrated out unless explicitly stated otherwise. The operators in are ordered naturally by dimensional analysis. Only the four-technifermion operators of the form ~L ~ can be represented by nonderivative terms, i.e. AABCDUBCUI~A. A familiar implication is that only four-technifermion operators of the form ~L can influence PGB masses (to first order) [201.To find vertices containing the Z field we must look at the next order in the derivative expansion of the perturbations to the phenomenological lagrangian. The twocovariant-derivative operators which transform like possible four-Fermi terms (when gauge couplings are turned off) are

BABCD(D~U)BC(D~Ut)DA,

CABCDUBCU~A~Tr[(D~U)(D~Ut)],

DABCD(UD~Ut)BA(UtD~U)DC, FABcD[(D~U)(DMUt)]

EABCDUBC[Ut(D~U)(D~Ut)1DA +

BA6DC’

RABCD(UtD~U)BA(UtD~’U)DC,

c.c.

GABcD~B/i[(D~Ut)(D~U)]DC’ LABCD(UD~Ut)BA(UD~Ut)DC.

(3.2)

With vanishing gauge couplings, the operators with coefficients AABCD 6ABCD transform as ~ and those with coefficients RABCD(LABCD) transform as —

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J~ J~(J~ -Jr). The ordering of indices in eq. (3.2) reflects these transformation properties. It is assumed that potential contributions to BABCD or EABCD which can be included into FABCD or GABCD are so included. However, it will be convenient at times to work with BABCD

=

BABCD

+

FABCC.~CD+ ~ABGAACD,

(3.3)

instead of B, F, and G separately. There may be relations among the coefficients. Note, for example, that if the are nonzero but the ~‘A~BCD are identically zero, then the coefficients AABCD 6ABCD are proportional to ‘A~jCD as tensors and hence to each other; (this follows from standard arguments used in spurion analysis, see e.g. refs. [35,36]). However, if both ,cd’A~,jCD and cIIA~CD are nonzero, then the tensor coefficients AABCD GABCD apparently may be different linear combinations of the tensors ~ç 1f(t) and ~(2) We will assume below that this is the case, so that in general A/iBCD GABCD are not proportional to each other as tensors. A definitive statement on this matter requires more detailed knowledge of derivative expansions than is available at present. In any event, all proportionality constants are real because technicolor interactions are CF conserving. By gauge invariance of the fundamental lagrangian, only gauge-invariant combinations of the above operators can occur in the phenomenological lagrangian. The coefficients of the operators in eq. (3.2) can be expanded conveniently in the form —

—

—

A

—

am~j3mpym~~~m5 —

‘IT/Ti

a,~ ~

a,~Tm,,,n~Tmym8~

B amj3rn0ym~8m~ —‘~“T’T~ If y/3 a8Tm,,rn~Tmym~’ —

‘IT’TJ

—

am,j3m~ym~~m~ — Cjj ~

D 1~

Famj3mpym76m,,

ii1T~ y~ ~ ~,-1’JT

i

I

— —

i

I

e‘IT’T’ 1~ ~l3

L

a.3Tm,~m0Tm~m~,’

—f’IT’T~ I J1J y13 a~Tm,,mpTrnyin~~

—

amj3meytny~m,,

I j a8TmmTmm,

—

Gamj3m~ym7am~~IJ R

~

— —

y~ aaTm~,me1~1 tn~m~~ 1

r

T aaTm,~m0 ~

1~ ~ 3

~

=1’~T’ Ii y/ T~~

,n~m 0 ~

-

4

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Extended technicolor

similarly, Barn pm~ym ~ ~ Here the expanded notation of sect. 2 has been restored, e.g. AABCD corresponds to Aam~m~ym~mThe 5. sup=

pressed summations run as I, J 0,.. ., N~ 1 and i, j 0,. . - , 3. Also, the T’ are a basis for all hermitean ND X ND matrices, and the T’ are a basis for all hermitean 2 x 2 matrices. We will choose particular bases in sect. 5. Hermiticity of the lagrangian requires =

ajj =aJ/, 11* Ii

=

—

b~*=b~,

H

JI’

=

If

d’~ JI’

=

3 b~~= 1, ~~-~i*—

iii

—if!

ii iI* ii Jij* 1ii g11iI* gj1, r1~ r~1, ~ 2)L invariance restricts the allowed values of the expansion coefficients so that SU( a0~,b°~,b°’,c01, d°~,e°~, only fOI, gO], ~ i~ i~,100 can be nonzero. Hypercharge invariance further restricts some of these coefficients but it is convenient not to explicitly state these conditions at this point. We conclude this section by noting that under a CP transformation the operators given transform by having A,.. . ,L, R, replaced by A*,.. - , L*, R*, aside from the usual effect on space-time coordinates. In technicolor theories weak CP violation should arise from CP-violating phases in four-Fermi operators connecting ordinary fermions to technifermions [37]. It is natural to expect that there would be CP-violating phases in four-technifermion operators as well. The size of such effects is difficult to constrain precisely so we allow for arbitrary CF-violating phases in our analysis of ETC interactions. =

=

=

=

122

=

4. The chiral symmetry breaking scale The phenomenological lagrangian approach is useful at energies sufficiently low that the composite (pseudo-)Goldstone bosons can be treated efficiently using elementary fields. In practice, this means that leading-order results from phenomenological lagrangians are subject to corrections which scale as powers (perhaps modified by logarithms) of ratios of an appropriate kinematical energy or mass scale divided by the chiral symmetry breaking scale, ACSB, a scale characteristic of the theory. For example, higher-dimension operators occurring in the expansion indicated in eq. (2.2) may lead to such corrections. The added dimensions arise from additional covariant derivatives or gauge field strengths, and are scaled by ACsB to an appropriate power. Analogous comments apply to the description of first-order ETC-induced effects in so that for example the tensor coefficients B—G of eq. (3.2) are expected to be of order O(A/A~ 5~). Consequently, the vertices we are interested in are inversely proportional to A~sB, and it is important to understand the limits on its magnitude.

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At present it is not possible to calculate ACSB. The accuracy of phenomenological lagrangian calculations in describing the properties of the pseudoscalar octet suggests that in QCD, A~5~ 1 GeV. A naive rescaling suggests that in technicolor ACSB is of order 1 to a few TeV. However, the parameters ND and NTC may be significantly different from their analogs in QCD, so it is relevant to consider the dependence of Ac5~on them. There is one approach which gives an approximate upper bound on ~ It is motivated by the perspective advocated by Weinberg in ref. [161.Most of the basic notions appear in ref. [38]; the complete argument in a more general form appears in refs. [35,39]. Roughly, it uses the principle that coupling constants should be stable under reasonable changes of the renormalization scale. The derived approximate bound is ACSB 4irf in the case of SU(2)>< SU(2) chiral symmetry. The argument of refs. [35,39] is reviewed in more detail in appendix A, and is extended to the case of SU(2ND) x SU(2N0) chiral symmetry with the resulting approximate bound, ~ <4irf/~/~~.Although the reasoning is self-consistent, the result should be regarded as somewhat tentative because we can give no independent physical argument for the suggested decrease of A~50with increasing ND. The calculations of appendix A follow ref. [40], and are applicable to both —~Th and to In QCD the bound ACSB ~ 4irf apparently is nearly saturated. However, in general this need not be the case. In a conventional large NTC approximation in technicolor (or N~in QCD), ACSB is 0(1) while f is O(1/~~(or %/7~))[40,41]. (In a realistic technicolor theory ATC ~ 1/~,/~ so that f is 0(1) and 7~L~. This theconsistent W mass iswith fixed.) in since the large ACSB -~f/~/ is the Consequently, previous bound largeNTC NTClimit, arguments are applicable only for ND NT(~. As a final consideration, eq. (2.5) gives f~v/~/i~where 1) iS a fixed scale. These three considerations all act to suggest that ACSB could be rather small if there are several technifermion flavors: ACSB

~

(4.1)

(4~-L)/ND.

Recall that a generation of technifermions has

ND

=

4.

5. PGBs and their related symmetries Here we form a basis for the generators of the SU(2ND)-valued field U(x), where a, /3,... 1,..., ND label the technifermion doublets and m~,m 0,... 1,2 denote their isospin components. Let T’, I 0, ..., N~ 1 be a basis for all hermitean ND X ND matrices. Choose T° to be V2/ND ‘NDXND~ For each pair a > /3 define two Ts, T’ and T’ =

=

=

—

M. Soldate, R. Sundrum

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Extended technicolor

say, as =

‘3a

=

—

=

1,

T~’~i, =

with other elements 0.

(5.1)

Then define the remaining Ts to be the diagonal, traceless matrices /2

V

k(k+1) diag(1,1,...,1,—k,0,0,...,0).

The Ts are normalized to Tr(T’T~)=~ Now if we let T’, i 0,1,2,3, be ‘2x2 2ND >< ~ matrices T1T’ form a basis for and the Pauli matrices respectively, the the generators of SU(2ND), except for T°T°. Thus for U(x) in the neighborhood of the identity, we can write, =

3

U(x)=exp i ~I~°(x)T°r~ j=1

/ /(%/~f)exp /

Nd—I

i

3

~

~J~.~(x)T~T1 J=1 j~0

/ /(~f) / (5.2)

where I-~~(x) is the hermitean field operator of the PGB having the quantum numbers of the bilinear ~(x)y 5T~T~~J(x)

=

(5.3)

~am)Y5T~Tm~mp(X)

As discussed in appendix B, Landau gauge is a convenient gauge choice: there is no mixing of PGBs and either the three true Goldstone fields (f~°) or the W and Z themselves, both at tree level and at one loop. This is not the case in the obvious analogs of other RE gauges (and unitary gauge). Since we work in Landau gauge the unphysical Goldstone fields are present, and should 2 maintain by 0 in theunitarity usual way. canceling the unphysical gauge propagator pole effects at k We will divide the physical PGBs into two categories: A in which [Ti, Y] 0, =

=

and B in which [Ti, Y] * 0. In category A, the fields P~(x) and P~(x) are electrically neutral, while P~(x) 1/v~[Pj’(x) ~ iP 1(x)] has charge ±1. The action of CF on category A is simple. If T~is real 2 =

CP: P~ 3(x)

—P~3(~), P~,_(x)

—P~~(~),

(5.4)

M. Soldate, R. Sundrum

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Extended technicolor

13

while if T~is purely imaginary CP:Pg3(x)—~PJ3(i),

~

(5.5)

where i is the parity transform of x. These two types of PGBs will be referred to as CP odds and CF evens respectively. The possibility of CP-even, neutral composite Goldstone bosons is familiar from the kaon system. Category B contains PGBs with zero or unit charge if there are doublets of technifermions whose hypercharges differ by two or four. Given that these are unconventional assignments and they do not occur in n-family models, we will comment only briefly on vertices containing such possible PGBs in sect. 11. Until then all PGBs discussed will be assumed to belong to category A.

6. Vertices of interest The phenomenological operators of lowest dimension which can describe the decay of a (possibly off-shell) Z into one or two neutral PGBs or one charged PGB are Z~ZP~PO~3, ~ ~ z~Fr3~P3L, Z,~W’~ P~.Because they have the lowest dimension, if they occur in the phenomenological lagrangian they are expected to dominate the contribution to the decays mentioned. In ref. [24] the absence in the leading term of of the single PGB couplings was shown in general, while the absence of the Z~,P~9’~P’ couplings (for neutral PGBs) was shown in the one-family model. We will now show that in fact none of the ZJ~Z’~P vertices, and none of the Z~P3’~P’ vertices involving exclusively category-A neutral PGBs, can occur in the phenomenological lagrangian at all unless ETC effects are included. However, it is possible for Z~P9~P’ vertices involving category-B neutral PGBs to occur in the dominant term of the phenomenological lagrangian in the absence of ETC effects; the amplitudes for these vertices are given in sect. 11. The reasoning below is also valid for vertices obtained from those above by arbitrarily inserting derivatives and contracting Lorentz indices in any way, so long as the e tensor is not used. Even if this tensor is used, the argument below for CP evens remains valid. First consider all the vertices above involving neutral PGBs which are exclusively CF odds. It is easy to see that these vertices are CF odd. Since the fundamental lagrangian is clearly CP conserving in the absence of ETC interactions, the corresponding phenomenological theory must also be. So these vertices cannot occur in the absence of ETC. For Z,~Z~P vertices involving neutral CF-even PGBs, the important observation is that the fundamental theory without four-Fermi interactions possesses global “doublet-parity” symmetries, whereby the left- and right-handed fields of any doublet can be multiplied by 1, leaving the lagrangian invariant. Since these —

14

M. Soldate, R. Sundrum

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Extended technicolor

symmetries are vector symmetries they are unbroken by technicolor chiral symmetry breaking. Because CF-even PGBs necessarily are mixtures of two different doublets, the ZJ~Z’~P and Z~W,~Pvertices involving them are not invariant under these symmetries and cannot occur in the phenomenological lagrangian when ETC interactions are neglected. For those Z~Pa~P’ vertices involving one category-A CP-even neutral PGB and one category-A CP-odd neutral PGB, so that the vertices are CF conserving, the symmetry violated is an SU(2) vector symmetry which mixes one doublet with another doublet with the same hypercharge assignment. Such symmetries are respected by the fundamental lagrangian in the absence of ETC effects. The “constituent” techniquarks of the CP-even must belong to doublets with identical hypercharge in order for the PGB to be in category A. Referring to sect. 5 it is clear that the CP-even must transform like the second component of a triplet representation of the particular SU(2) which mixes these doublets. The CF-odd however, can in general only transform like a linear combination of the first and third components of the triplet representation, members of the doublet representation, and the singlet representation. The vertices therefore cannot be part of an SU(2) invariant, so they cannot occur in a theory invariant under the symmetry. The argument cannot be used for vertices containing PGBs in category B because the relevant SU(2) symmetries are explicitly broken in the fundamental theory by the differing hypercharge assignments of the “constituent” techniquarks. As mentioned above, such category B vertices can occur in the leading term of the phenomenological lagrangian when ETC effects are neglected. The vertices iZ~(W~/F~‘V~Fi~) can occur without incorporating ETC effects. They do not occur, except for K 0, in the leading term of eq. (2.2) as explained in appendix C. However, subleading terms may give rise to such vertices. For Tr[Y2D 2DAUtDAUI can occur example, the operator g’2B~”B,~,. + (Y+ T3)field strength for the in -~h even without ETC interactions;5UDAUt here B~ is the hypercharge gauge field. It should be suppressed by a coefficient of order f2/M~~~ because of the two additional hypercharge field strengths. (Both terms in the trace are required in order to preserve the symmetry of parity augmented by the interchange of left- and right-coupling matrices, as present in the fundamental lagrangian.) Upon contracting the hypercharge gauge fields, one can extract the vertex iZ~(W~P~ — W~P+K)from here. Using the chiral symmetry breaking scale as the cutoff for the momentum integral gives amplitudes of order (g’2/(4~r)2~/7~)gg’sinO~v. These amplitudes are much smaller than those for the production of say a single neutral Higgs particle. They are phenomenologically negligible. We show later that all the vertices listed above can occur in the ETC-induced perturbation to the phenomenological lagrangian with sizable coefficients. One may wonder whether in the absence of ETC effects one may nevertheless find vertices of the form Z,~Z~F(~< higher 3 up in the derivative expansion where the —

=

M. So/date, R. Sundrum

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Extended technicolor

15

Lorentz indices are contracted with an tensor and partial derivatives acting on the particle fields. This possibility is ruled out for the CP evens because of doublet-parity conservation; however, for CP odds which are doublet-parity even, vertices such as PBP3~ZV3BZI3P~~ which 3, do not violate CP, are possible. Such forms of vertices occur for example in 2[UtD2U_ ~

(D2U)Ut]

—

(D2Ut)U]};

(Y+T3)

(see ref. [35] for the analog in ~ —~ yy). This operator should be suppressed by a coefficient of order f2/fl~sB in the phenomenological lagrangian. (Again the last two terms are required to preserve the augmented parity symmetry of the fundamental lagrangian.) They cannot occur lower than this in the derivative expansion however, so there are two extra contracted partial derivatives in the vertex. (There are other contributing operators at the same order of the derivative expansion though.) The amplitudes are at most of order (1/ ~ where m here is the mass of the PGB in the vertex. Thus for PGBs which are not too heavy, their actual masses suppress the vertex, unlike the amplitudes for the vertices induced by the four-Fermi interactions. In this case the production rates for PGBs from such vertices are negligible in Z decays. Another contribution to vertices of the form ~va~ ZVaBZPF()!c 3 comes from the Wess—Zumino term of the phenomenological lagrangian. The contributions these make to the production of a neutral PGB are at most very small compared to the rate due to a Higgs-like vertex for any models with moderate values of PGB mass, NTC, and ND, and under these circumstances they are phenomenologically negligible in Z decays [251. It should be noticed that the vertices from ~z~h contributing to the production of a neutral PGB have a different tensor structure than the Higgs couplings and those lowest dimension operators we will find in the ETC-induced perturbation to the lagrangian, and therefore in principle their effects would be experimentally distinguishable from the latter even if total rates were comparable [42]. The E tensor cannot be used in the vertices with two PGBs and one Z because there will always be at least two commuting partial derivatives contracted with the antisymmetric tensor.

7. Strength of vertices in Now that the explicit form of is known in terms of the gauge bosons and PGBs, the coefficients of the vertices we are seeking can be extracted. A contribution to the category-A vertex, Z~Z’~PO”, will be worked out below; the full results for the category-A vertices are collected in table 1. The coefficients in the table labeled as Z~’P3~P’ are equivalently the coefficients of

16

M So/date, R. Sundrum

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Extended technicolor

TABLE I ETC-induced couplings ofcategory-A PGBs

Vertex

Coefficient in 00

00

00\

2)/f]Im(b~~ + cOK + eOK — eKO)

Z~,ZI~P(

2~/~[(g2+g’

03

03

Z 2~/~~[(g2 + g~2)/f]Im(b~ + c~ + eOK

0Z’~P~<

03

z0p~
_8(%Ig2+g~2/f2)Im(b~

ZEP~a.~Pf

_8(~/g2+g2/f2)lm(b~+e~~

ZaP~P~

_8(~g2 +g’

03

/f2)Im(b~

03 —eLK)

0() + eKL

((0 eLK)

cos Ow(Im b~ — iRe

cos 0w Im(

30 r~K—

~)

r03 )+ ~ 0~ ~

OK

OK

cos O~Re( i°°— ~

—

eKO)

+eKL —eLK) (13

2

—

33 00’ + ig2 coso~Re(rOK— rOK) g2 Re(r~.—r—r~/<+r~/~)

2 cos O~Im

)+ g

e~)<]

F’a,~P).Consider the term b°~T’T~T&m 0T/n

(

ym0( D~U)0m0ym D~Ut)0moam,,

3 aO y1

1J

in

&z~h. The

coefficient of Z~Z~F(~ in this term is

( —1 b°~T’ Tc;’,~T~mTInm~ ~çii~j[LzT~~
—

t T~~
-1 + [i(L~

Rz)I 0mpymy~[LzTKTO

—

TKT0RZ1;moam~]

2 +g’2) {Tr(T!TK)Tr(TJ)Tr(r3r3r3) (g —

(g2+g’2)

f

Tr(T”)Tr(T~T’~)Tr(r3r~T3)}

(~-~)

=2~/~ (~2+~’2)J~(~oo)

f

(7.1)

M. So/date, R. Sundrum

where we have used

=

[TK,

Y]

=

Extended technicolor

[-(Im

17

0. From eq. (3.3),

[(Re g~) + (Ref~)~°]Tr({T~, TK},

b~+

+

/

g~)+

t0]fJKL,

(Imf~

TL)

(7.2)

so that Im(b~) Im(bg~). One can show easily that the coefficients appearing in the table multiply operators in which are hypercharge invariant. Thus hypercharge invariance does not constrain the above results in any way. Following the perspective of Weinberg [16], diagrams involving one PGB loop may make comparable contributions to PGB production. We have ignored these but do not expect them to alter the general phenomenological conclusions of this paper. One can see that the amplitudes for Z~Z~P~ appearing in the table depend entirely on coefficients of ETC-induced operators which are either CF violating for vertices involving a neutral CF odd, or are doublet-parity violating for vertices involving a CP even. Similarly, the amplitudes for Z~P3P’violate either CF or the vector SU(2) of two doublets with equal hypercharge. Of course this had to be so since these symmetries are violated by the vertices. As pointed out earlier, their preservation forbade the occurrence of the vertices in the absence of ETC interactions. These coefficients also violate the axial symmetries corresponding to the PGBs appearing in the vertex. This is necessary for the Z,~Z/~P~ coupling (for a category-A PGB) because the corresponding axial transformations, generated by T~r° and T~T3,are not explicitly broken by the Z gauge interactions and because this coupling is nonderivative. =

8. Masses and production amplitudes for the PGBs The neutral PGB vertices of the table receive contributions only from the operators proportional to B, C, D, and E of eqs. (3.2) and (3.3), which are related to the operator AABCDUBC(~DtA through the derivative expansion. Therefore, there is at least a qualitative connection between the strength of these vertices and the ETC-induced PGB masses. This is consistent with the estimate in the literature that e.g. the ETC-induced Zt’Z,~Pcoupling is O(m~,/A~~)g2u [11]. However, we will argue generally that the strength of a given Z’~Z~P coupling is not directly tied to the mass of the PGB appearing in the given coupling; that in first order increasing the strength of a particular perturbation which leads to a Z’~ZEF

18

M. So/date, R. Sundrum

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Extended technicolor

coupling in the table does not increase naturally the mass of the PGB appearing in the coupling. This is significant for the phenomenology of Z decays because it implies that PGBs light enough to be produced in Z decays can still have large couplings to the Z, reasonably. The general argument relies on symmetry considerations discussed previously. The Z~Z1~P couplings can only be induced by perturbations which are either CF odd or doublet-parity odd (or both). In first order such perturbations cannot affect the diagonal PGB mass terms in the phenomenological lagrangian since these are necessarily even. Therefore, the trace of the PGB mass matrix is not influenced; if some PGB mass eigenvalues are pushed up by increasing the perturbations’ strengths, others will be pushed down. The argument is also valid for those Z~F~F’vertices which are CF or doublet-parity odd, though the vertices which are even under the two symmetries can get contributions from the same fundamental operators which contribute to diagonal mass terms as will be illustrated later. We see that light PGBs can have strong ETC-induced couplings of the appropriate sort. 1,h influences the PGB mass matrix through operator AABCDUBCU!~A ~— theThe terms quadratic in the PGB in fields:

~~uad

=

-

~

1

[~

N~—1 L0 Tr(TL{TK,

T~))(Rea~%)- 8(Re a~)](P/PjK + P(/PQ~)

N~-1

+2

~

Tr(T’~{T’<,T~})(Rea~ 1) 8(Re a~) pj’P~ —

L=0

N~- I

L=I fLJK~~’O)

+2(Ima~)]P/PkKe)51].

(8.1)

The suppressed sums run as J, K 1, - . - , N~ — 1;not j, acquire k, I 1,2,3. of 9, do mass Because from ETC SU(2)L invariance, the certainly true Goldstones, P~ interactions. There are enough parameters to give all other pJ 5 (J * 0) mass. For example, the a~-perturbationalone gives a diagonal PGB mass matrix 2)agg. Some further remarks on the a~-perwith equal diagonal elements, —(8/f turbation are contained in appendix D. Note that in general the mass matrix is not diagonal so that PGB mass eigenstates will be linear combinations of the P/. =

=

M. Soldate, R. Sundrum

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Extended technicolor

19

The sum of the eigenvalues of the ETC-induced PGB mass matrix, M2, can be computed by calculating its trace, 32

Tr(M2)

=

~

N~—1 L~I

a0L0~— agg(N~— 1)

(8.2)

-

Vacuum stability requires that the eigenvalues must all be positive. The average mass squared of the 4(N~ 1) PGBs (not including the true Goldstones) is thus —

8 ~

1

~00=

f

N2—1

E

.

(8.3)

D

The general argument given above can be illustrated in a particular case. Suppose that the coefficients ~‘ABCD are nonzero but the ~~A~CD are identically zero. Then as discussed in sect. 3, the coefficients AABCD GABCD and hence the expansion coefficients a~< g~< of eq. (3.4) are proportional to each other as tensors with real proportionality constants. One can check that the Z~Z~F couplings and the CF- or doublet-parity-odd ZILF0’~F’ couplings of the table depend on parameters which only appear in off-diagonal entries of the ETCinduced PGB mass matrix. In fact, the Z~Z~P~ couplings are all proportional to Im(a~), parameters which do not even affect the PGB mass matrix itself. In contrast, the couplings iZ~(W~F~— W~P~)are CF and doublet-parity conserving for TK diagonal; these symmetries are also conserved by some of the Z~,Pa’-tP’ couplings. Using the definition of b~’J and the proportionality conditions, these couplings in general receive contributions proportional to Re(a~), parameters which do influence diagonal elements of the PGB mass matrix. The coupling strengths of the table, aside from those for iZ~(’V~/F~W~Pf~) and the CF and doublet-parity-conserving Z~F0’~P’, are not related directly to the mass(es) of the PGB(s) appearing in a given vertex. Instead, we will express the coupling strengths in terms of m2 which then serves as a measure of the average strength of ETC perturbations on technicolor dynamics. This could be misleading if, for example, ETC interactions are approximately CF and/or doublet-parity conserving. It is natural to assume that the x 1~, {1/J, c1~,d,~,e1~}coefficients are all of the same order since they are induced by the same fundamental operators, 2and are in obtained the same order of the derivative expansion*. Using the equation for m above and the fact that v2 (the approximate nature of this relation being —

—

—

=

NDf2

*

One may have wondered earlier why the factor of l/2ND was included in the operators UB(U~A(1/2ND)Tr[DOUD0UO] multiplied by the coefficients C of eq. (3.2) in the phenomenological lagrangian. The point is that with this definition, the ancestor of the above operator in the derivative expansion is UfiCU~A(I/2ND)Tr[UUt]. Performing the trace gives a factor 2N 2ND, leaving the operator whose coefficient in -~~his just A, which is the 0 ancestor which cancels for the other the I/two-derivative operators in Thus it is only with this normalization for C that we can expect that c 1~is of the same order as the other expansion coefficients b1~,d1~ gjj.

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M. Soldate, R. Sundrum

20

Extended technicolor

explained in sect. 10), we find that the ZILZ,2PK amplitudes are of order 2+g’2)~v ImxOK a a 00 _~~_

—

(g

where from the table it is clear that the strength can be several times larger if all the possible terms contributing0~.Similarly to the strength do not cancel By XOK the strengths of the much. Z,.~PKa~~PI~ are we of mean a typical size for the x0 order /

2

V(g +g’

2

)

ImxKL

a

_~~_

a ~m 00

2 -

As mentioned earlier, the true PGB mass eigenstates will be linear combinations of the pK, but the above estimates apply equally well to the couplings of these physical particles. The Z~W’~F cannot be estimated in this way because of the presence of 1 and r coefficients which are unrelated to PGB masses. Later we will comment on these vertices and provide reasonable guesses for the above ratios of coefficients. We emphasize that if we were working within a specific ETC model we could say more possibly about the vertex strengths by using the details of the table of vertex amplitudes. Actually, the couplings ~ of particular phenomenological interest, have an additional complication. These couplings are inevitably associated with vacuum stability issues because they contribute to F’ tadpoles after contracting the Zs. Vacuum stability requires that there be no net tadpole for any field. The tadpole contributions from the operator AABCDUBCUJ~A are (8~/7~/f)(Ima~)P/. If g g’ 0, then in the stable vacuum Im a~/, 0. In the simple case where only the coefficients dA~jCD are nonzero and the ZEZI~FJcouplings are proportional to Im a~, only, this vacuum (where Tm ag~ 0) is stable even when g and g’ * 0 because there are no nonzero Z~Z~P’ in it. In the more 2)(or areL~I~W’~P~) nonzero, couplings this vacuum is not stable in general case where both ~ç1fW and ~çj( general; there may be nonzero Z,~Z’~P’ couplings which lead to tadpoles of order —

=

=

=

=

(fl~SB/(4~)2)~/~ç(g2/f)(Im x~,). The vanishing of all net tadpoles requires cancellation of these terms by terms from AABCDUBCU~A which would arise through a vacuum reorientation of magnitude proportional to aweak. Thus, nonzero ZEZ~P~ couplings should be allowed if both coefficients ~1ABCD and ~‘ABCD are nonzero. 9. Some estimates in the absence of a specific model In this section we explore the possibilities for the vertex strengths in theories where ND is moderately large. The ordering of indices used to define the expansion coefficients in eq. (3.4) is convenient for the coupling calculations;

M. Soldate, R. Sundrum

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Extended technicolor

21

however, the indexing of the generators T’ and T’ is related to the natural ordering in ETC by a Fierz transformation. This makes it more difficult to relate models of ETC interactions to PGB couplings without detailed understanding of ETC interactions. We will resort to the rough estimates below. We first try the assumption that there are many massive ETC gauge bosons at the ETC scale, which connect the technifermions without much regard to their flavors. This might be the case if the ETC theory contained large multiplets of technifermions. This scenario can be modeled by taking each AamOm0ym6m0 to be of the same order, and similarly for X

=

B

G. Using the fact that

‘J±TITJ

If

16

fry

I

V

~aTmpm~,Tm5m~

am~jim0ym~~Jn0

we find that the x~ are all roughly of the same order. In order to estimate coefficients with any 0 doublet indices we have used

E~

x (numbers of order 1)

(9.1)

a,,B

since this is a reasonable expectation if the signs of the ND summands are not greatly correlated. We will denote this possibility by (i). It is however possible (ii) that if the dynamics somehow prevent cancellation, x (numbers of order 1)

ND.

(9.2)

a,f3

In this case x00 will be of order ND XJf and x~ will be of order ~ I, J * 2. 0. These two possibilities give the following two sets of relations among the xs and m (i)

x 03 is of order 1

(ii)

x°~ is of order

0~

is of order

f2m2

for all I, J;

2

f 2m~ QP~T

A2 (JLVDJICSB

f2m2

8~A2csB

f 2m2 x 0~is of order 2 8A~~ 0

for I, J* 0. 11

(9.3)

M. So/date, R. Sundrum

22

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Extended technicolor

In the above we have used the natural expectation that a00 and a~gare of order and X~JA~sBrespectively because the latter multiply operators induced by the same fundamental operator as induce the former but they are in one higher order in the derivative expansion. Another possibility (iii), is that ETC interactions do not significantly connect many of the technifermions. In this case though, it is much more difficult in the absence of a specific model to guess the relative sizes of the various coefficients. It is however hard to see how one would naturally avoid the inequalities,

1°t1A~SB

f 2m2 XJf

‘~

Qfl2 °

for all I, J.

,

CSB

The reasoning assumes that there is no great cancellation between a°° and a~.If this assumption is false then m2 is much smaller, and consequently the x 1, itare 2. This possibility will be referred to as (iv) though is much larger in terms of m perhaps “unnatural”. Finally, the hierarchies in quark and lepton masses could be reflected in giving rise to large variations in the expansion coefficients. This possibility is difficult to explore without a model. 10. Masses of the electroweak gauge bosons The shifts in the electroweak gauge boson masses due to ETC interactions are given by =

2N~(g2 +gf2)[b~ +

r 00 00 =

c~+ 2(Re e~) L

33

r00

—

/00

11

22

r00

—

r00

-~-

~33 —

2N~g2[b~+ c~+ 2(Re e~j) (10.1)

+r~—r~+1~—1~} -

(Competing one-loop contributions arising from PGB mass effects have been studied in ref. [43].) Note that if custodial SU(2) symmetry is enforced, r~ r~ r~and the p-parameter is 1, whereas if the r coefficients make any significant contribution to the electroweak boson masses and do not maintain custodial symmetry, then there is the danger of unacceptable violations of the p relation =

[44].

=

0 0 ~f2?n2/8A2csB and assuming the r and / coeffiUsing our earlier estimate x0 0 cients are not much larger than the xs, we find that the masses of the gauge bosons are shifted from the value they obtain from the dominant term of ‘~h by a fraction of order m2/A~SB;a few additional related remarks are contained in appendix D. If m2 A~ 50,this modifies the exact relationship in the absence of

/ Extended technicolor 23 ETC interactions, V2 N~f2, to an approximate one, v2 NDf2, though one should nevertheless be able to use the latter in order of magnitude estimates. If the r or 1 coefficients are too large they will dominate the contributions to the electroweak boson masses and thus set the weak scale rather than the weak scale being set by ~ We do not consider this possibility since it violates the usual assumption that the weak scale is determined by technicolor interactions and not by ETC interactions, and also because, as mentioned above, large violations of the p relation can arise. Thus if m2/A~SBis of 0(1), the coefficients r and 1 are limited to be of the same order as the coefficients b~,c~j, and e~,while if m2/A~SBis much smaller than 1, the above mass shifts are all small compared to the contribution from the dominant term of ~h, provided that the r and / coefficients are not much larger than the x coefficients. It is possible of course that the dynamics of the theory are such that the various terms contribution to the electroweak boson mass shifts cancel to a great extent even though individually they may be large. The phenomenological lagrangian cannot inform us if this is so. In any case, the ratio m2/A~SBcertainly can be of 0(1) without unduly shifting the W and Z masses; if such cancellations occur in eq. (10.1) (compare ref. [45]) then it is acceptable phenomenologically for the ratio m2/A~SBto be larger than 0(1). M. Soldate, R. Sundrum =

11. Category-B PGBs We return to the possibility that there may be some category-B PGBs with unit or zero charges, thus capable of being produced from a Z along with a Z or W. As mentioned earlier this can only happen if there are doublets whose left-handed hypercharges differ by 2 or 4; (this does not occur in any conventional n-family model of technifermions). For category-B PGBs, doublet-parity conservation forbids the vertices Z~Z’~F and Z~W~Pin the absence of ETC. However, the neutral PGB couplings Z/LP9~F~ which are doublet-parity even do occur in the leading term of eq. (2.2), the relevant terms being in sin 6w(I~9.~F~/E)Tr([TJ, TK]Y) +~(F’a~P~’)(gcosow_g’sinew)Tr([~.,Tk]T 3)}. Specializing to the case of two left-handed doublets whose hypercharges differ by 2, the two electrically neutral fields are (1/V~)(P~ + P~) and 4~ (1/ %/~)(P~ — P~).The Z—q~0—4~ coupling is =

0w ~ =

2cos

=

—

~

(11.1)

M. So/date, R. Sundrum

24

/

Extended technicolor

This parallels the results of ref. [24] since ‘/~ and 4~have 131 1, where I generate the vector component of SU(2)L. The amplitudes for the ETC-induced vertices are similar to those for category A, the differences being in the indices of the expansion coefficients entering the amplitudes and the fact that the neutral PGB vertices can now depend on the d, I, and r coefficients. One implication is that the absence of tadpoles in the stable vacuum no longer forbids the Z~Z~F~ coupling under the circumstances discussed in sect. 8. For the purposes of our crude estimates of the magnitudes of these vertices, the differences between categories A and B will be of no consequence. =

12. Putting back color Up till now the effects of some of the technifermions being colored has been ignored. At this stage it is simple to explain the effects of QCD on our analysis. Some of the PGBs may not be color singlets. QCD effects raise their masses [20] and they are not expected to be the first PGBs seen. Some linear combinations of the PGBs (as we defined the PGB degrees of freedom) will be color singlets and unaffected by QCD. The~amplitudes for their vertices are proportional to simple sums of amplitudes of the forms we have given, and give the same results (for example, the amplitude for production of a neutral uncolored PGB made up of a color triplet of technifermions and a color antitriplet of technifermions isjust 1/ %I~ times the sum of the amplitudes for the production of the corresponding red—antired, green—antigreen and blue—antiblue PGBs, which are all the same anyway by color invariance). It should be noted that a techniquark triplet counts as three in the count yielding ND. 13. Putting things together Let us first summarize what we have found for the couplings of category-A PGBs. As discussed in the literature and in sect. 6 the non-ETC-induced couplings of one or two PGBs to the Z are negligible as long as the derivative expansion does not break down, with the exception of the coupling iZ~P~3~P. The ETC-induced couplings of interest have been tabulated in terms of parameters defined in eq. (3.4). With m2 given by eq. (8.3), we can expect for the neutral PGBs that the Z~Z~FK coupling is of order L~’

ImxOK _~~_

-ta

—

a 00

while the

Z,~PKi3~FL

coupling is expected to be of order ImxKL 2. a_oo_ a 00m 00 —

~(g2+g~2)

M. Soldate, R. Sundrum

/

Extended technicolor

25

If our guess that the XJf are all of the same order is correct, and are as given in eq. (9.3)(i), these couplings become simply

and CSB

~/(g2+g~2)

~

CSB

respectively. If m2/A~5~ is not too much less than 1, as discussed in sect. 10 the r and 1 coefficients ought to be at most of the order of the x coefficients. If this is true the Z~W~P÷ vertices in &J~h have amplitudes of roughly the same order as the Z,~Z~P vertices. The overall magnitudes of Zthe ETC-induced couplings are then tied to the ratio 2/A~SB.The couplings for m 11Z’~Pand Z’~W~P~ can be compared to the typical 2 +g’2)v/4, while the coupling within Higgs theories for Z,~Z~HQ, which is (g couplings for Z~FÔ’~F’ can be compared to that for Z,~H 2. 0a’~H6, namely ~g2 + g’ Therefore, if m2/A~SBapproaches 1, neutral PGBs can have couplings which are Higgs-like in form and strength, and charged PGBs can couple appreciably as Z’~H/,P~. From the largely phenomenological arguments of sect. 10, the ratio m2/A~SB can be 0(1) without unduly shifting the W and Z masses, and could be larger if significant cancellation in eq. (10.1) occurs due to special dynamical considerations. The ratio could be of 0(1) reasonably if both and AC 5B are each a few hundred GeV. Relative to conventional expectations, this requires that be large and AcSBbe2/A~~~ small. being of 0(1) would make the phenomenologiOne might argue that m cal lagrangian unreliable, since it is the typical suppression factor of the derivative expansion. However, it should be remembered that if we are looking at the amplitudes for the lighter of the PGBs, their masses may be significantly smaller than ~ For these light PGBs the derivative expansion is still good. The requirements to induce Higgs-like couplings may be met relatively naturally. Eq. (4.1) indicates that Ac 5~~ (3 TeV)/ND, so that ND about 10 would give a suitably small AC5B. Large ND is also necessary for “walking” technicolor [31,32], where is arguably large — conceivably several hundred 6eV. On the other hand, if ND is small, ~ is probably on the order of 1—3 TeV and the ETC-induced couplings are correspondingly small. In addition, if a neutral PGB, P0, with Higgs-like couplings is to be found in Z decays it must be relatively light. Such a PGB would require there to be significant splittings in the ETC-induced PGB masses; a reasonable situation might be m~0 30 GeV and 300 6eV. Such splittings may be more likely if ND is large for there are then many PGBs, and the PGB mass matrix depends on many parameters (see eq. (8.1)) so that appreciable cancellations in the calculation of mass eigenvalues are more likely. In this context it should be remarked that the

26

M. So/date, R. Sundnim

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Extended technicolor

lightest PGB is most likely to be found first experimentally. Our analysis strongly suggests that if a light neutral PGB with Higgs-like couplings is found in Z decays, then there should exist additional heavy PGBs to be found at higher energies. Under these circumstances one also expects to find significant Z’~W~F~ couplings which could then serve as a phenomenological signature for technicolor theories since these couplings rarely have analogs in standard theories with fundamental Higgs scalars. The above general results do not rely specifically on the estimates of eq. (9.3). The estimates given in this section are representative, given to illustrate that ETC-induced PGB couplings should be small if ETC-induced PGB masses are small but that these couplings can be large provided that PGB masses are strongly perturbed by ETC interactions. As discussed in sect. 11, the situation for category-B PGBs is similar to the situation for category-A PGBs. The primary exception is that the neutral PGB couplings Z~P0’~F’ can occur in the leading term of the phenomenological Iagrangian, with size of order g/cos Ow’ a Higgs-like magnitude. To conclude, as long as it is possible that ETC effects can perturb significantly technicolor dynamics, the couplings of the PGBs arising in the context of technicolor may be rather close to the couplings commonly associated with fundamental Higgs scalars. The resulting ambiguities in the interpretation of data can be relevant conceivably to results from Z decays. In the future, the ambiguities will become more severe as higher mass scales are probed. We have benefitted greatly from several conversations with T. Appelquist and L.C.R. Wijewardhana, and thank H. Georgi for helpful criticisms.

Appendix A REMARKS ON THE CHIRAL SYMMETRY BREAKING SCALE

Here we review the arguments of refs. [35, 38,39], reapplying them to the case at hand. The calculations neglect standard-model gauge interactions. The analysis is most easily described first when there are no sources of explicit chiral symmetry breaking, i.e. no ETC-induced perturbations. S-matrix elements of Goldstone bosons can be calculated in principle from the underlying strongly coupled dynamics. On the basis of a power-counting argument and qualitative considerations of unitarity, analyticity, etc., Weinberg [16] argued for an unproven but quite reasonable theorem that these S-matrix elements (and perhaps Green’s functions) in a low-energy expansion should be reproduced from the most general Goldstone boson lagrangian consistent with the assumed symmetries. The (local) divergences which arise perturbatively can be eliminated by

M. Soldate, R. Sundnun

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27

renormalizations of the couplings of local interactions in the Goldstone boson lagrangian with the introduction of a renormalization point, ~. It is reasonable that the renormalized interactions of the Goldstone boson lagrangian (and ultimately that the final corresponding local interactions given by the S-matrix) should be no smaller than the variation of the counterterms of the Goldstone boson lagrangian under a rescaling ~ by say e. More concretely, consider the one-loop counterterms arising from the interactions governed by the symmetric two-derivative term in the Goldstone boson lagrangian. The logarithmic divergences for SU(2ND) X SU(2ND) chiral symmetry at one-loop in this situation have been computed in ref. [401 using dimensional regularization.

f2

1

1

ND

ZI’ 4~p=

4A -“

2

fddx ~

(4lTf)

+ ~ND((3/~Ut)(a~U)(8~Ut)(aPU))

+

+

~K(a~Ut)(avU))K(3~Ut)(o~U))].

(A.1)

Here the brackets <...~denote a trace. In a regularization scheme more physical than dimensional regularization, the ultraviolet cutoff can be taken to be of the order of AC5B (the effective compositeness scale). These divergences are removed by renormalization of the appropriate four-derivative operators. For ND 1, stability under changes of the renormalization point jt suggests that A~5~ ~ 4~rf [35, 39]. Note that although O(N~)Goldstone bosons can circulate in the loop, the one-loop counterterms are proportional at most to ND. The one-loop results for arbitrary ND suggest that A~5~ ~ 4~f/~ Manohar and Georgi argued that 2,where the effective loop-expansion parameter for ND 1 is of order A~~/(4~f) AUV is the ultraviolet cutoff. Then with ~ Ac 5~~ 4~-f,the loop expansion is not obviously out of control and the bound Ac5~ 4irf is self-consistent. The ND dependence of higher-loop counterterms is not known, but large NTC arguments make it plausible that each additional Goldstone boson loop (corresponding roughly to an additional fermion loop) again leads to at most one factor of ND. Then for arbitrary effective loop-expansion parameter would be 2. AgainND it is the self-consistent for Ac NDA~v/(4~f) 5~~ 4~f/~ Actually, in this paper we are concerned with the derivative expansion for explicit breaking terms. This consideration does not appear to affect the previous conclusions. Using [40] on the perturbation AABCDUBCU~A 2K(xtU the+ techniques xUt)), the of logref. divergences (poles in 4 — d) in Z rather than ~f 1100~,linear =

=

M. Soldate, R. Sundnim

28

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Extended technicolor

in A/iBCD are

4—d ~

2AABCDfdx{ND[U(0U)(3~U)1BCUDA t/9/~U~ +NDUBC[Ut(a~U)(a.,~Ut)]DA + UBCUJ~AK3~U

+ 2(U3~~Ut)BA(Uta~U)DC— ~BA(~~-’

3,.~U)DC (a~uaMut)B/i~DC}.(A.2) —

Some of the one-loop counterterms again are proportional to ND, so the previous inequalities are expected to hold for the symmetry breaking terms as well. In these calculations the PGB masses have been treated as perturbations. If some PGBs are sufficiently heavy this may not be reasonable since they may decouple effectively. However, this complication is beyond the scope of these only suggestive calculations. Appendix B GAUGE CHOICE

In order to have a consistent perturbative treatment, it is desirable to work in a gauge in which the PGBs do not mix with either the W and Z or the true Goldstone bosons (F°s). To illustrate the potential difficulties, consider the simplified lagrangian =

~f2 Tr(D~UD~Ut) + BABCD(D,.~U)BC(D~Ut)DA.

(B.1)

Defining T3G~ 2cosOwT~

~(r;+T~~),

(B.2)

the mixing from eq. (B.1) bilinear in gauge fields and Fs is of the form (B.3) where =

(Re b

0~,)Tr(r~(T’, ~rk}) +

i(Im b~,)Tr(r~[r1,Tkl)

0

with summations j 1, 2, 3 and k, / 0,. . - , 3. It is somewhat unusual for there to be bilinear terms mixing gauge bosons and PGBs. =

=

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Extended technicolor

29

One obvious approach to eliminating the mixing of eq. (B.3) is to use a generalized R(gauge-fixing term, 2

a’~G~+ 2~ (i~°+p~~~) .

~

(B.4)

However, ~ gives pOpf mixing except in Landau gauge (~ oc); only in Landau gauge is the perturbative expansion straightforward to develop. Note that ~ 0 does not correspond to the usual unitary gauge because the mass of a linear combination of P° and pf5, not P° alone, is infinite. Even in Landau gauge it is =

=

possible that F°—F’mixing can appear, through ioop effects. For example, terms of the form (Z~ZEor ~~~01)PO_PJmay appear in the expansion of 4 in eq. (B.1). Contracting the two gauge bosons as a loop can give rise to F°—P’ mixing. The potential F°—F~ mixing in Landau gauge will be shown not to arise at one loop. If g’ 0, it is evident from the constraints of SU(2)L invariance that the p° fields, in the unitary matrix exp{i~...1P,°(x)T°T~/(V’~f)), cancel out of the perturbations B—G and L of eq. (3.2) directly, and also out of the perturbation R of (3.2) after integrating out the gauge bosons. g’ * 0, the F°(x)fields not cancel 2)LFor invariance. In the following do discussion we but their governed by SU( for simplicity. The coupling will work presence only withiscategory-A PGBs, =

—

gg’B~

E

~ (B.5)

arises from ..4. The primed sum runs only over category-A PGBs. The following coupling in =~ follows from eq. (B.5) and SU(2)L invariance (see eq. (C.2)): 8 g

~

[(Reb~3)(F(P~+FIP~])

+

~

(B.6)

f~=0

•

•

(a)

•

a

•

•___o

)b)

)c)

Fig. 1. Feynman diagrams describing one-loop contributions to

p0_pf

mixing.

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M. So/date, R. Sundrum

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Extended technicolor

There are three diagrams which can contribute to P°—F~mixing at one loop. Fig. la does not contribute in Landau gauge [24]. The contributions of figs. lb and ic, arising from vertices (B.6) and (B.5) + (C.l) respectively, cancel. We have not investigated these issues further; no additional difficulties are expected. Appendix C Z-W-P

COUPLINGS IN LEADING ORDER

Couplings of the form iZ’~(W,/F~ — W,~F~) are allowed by CF conservation to if under CF F~ (x) —F~+C~).By explicit calcuappear in ~f2 Tr(D~UD~Ut) lation the only such coupling present is [24] —~

i(f 11/2)gg’B~(w~P°± — W~P~),

(C.1)

where B~=A~cosO~+Z~sinOw and F2 =(1/~/~)(F iP~). The nonderivative coupling for true Goldstone bosons and the absence of couplings for J * 0 are at first sight surprising, but are accounted for by considerations. The 2)Lsymmetry under which nonzero coupling (C.1) is governed by global SU( 6I~°(x)=c~+O(P°), t~W~/(x) = _(2/fl/ )fJklkw1(x) (C.2) The constant shift of P° in eq. (C.2) links the W—Z—P° interaction to the W—Z mass matrix. If g’ = 0, custodial SU(2) is unbroken so that the gauge-boson mass matrix is of the form -~-m~W This 2. is itself invariant under global SU(2)L, so any 4 P°—(gaugeboson)2 coupling must vanish when g’ = 0 to preserve SU(2)L. Similarly, if g’ * 0, the W ± and Z are nondegenerate, and the coupling (C.l) is required to preserve global SU(2)L. The W—Z—F~~~°~ couplings are governed by different symmetries. If g’ = 0, there is a good global SU(ND) vector symmetry under which only p~~*O) transforms. This forbids any P(J*O)_(gauge boson)2 coupling. If g’ * 0, the SU(ND)V symmetry is explicitly broken, though only by terms [Y, U(x)]. Since i[Y, T’] is a sum of SU(ND) (not U(ND)) generators, this explicit breaking is not sufficient to allow a ~°~—(gaugeboson)2 coupling. Appendix D ENHANCEMENT OF WAND

Z

MASSES

This appendix contains a few remarks on the perturbation corresponding to a~, and its relation to the W and Z masses. In terms of technifermions the ETCinduced perturbation to the fundamental lagrangian corresponding to agg can be taken to be 1

_____ /7‘Lam~’/’Lpm~)PR~rnp’/p~’PRam , \j’7

~ ~‘oo — — ,t2

0) ETC

5.

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Using Dashen’s formula for the PGB masses and the vacuum insertion approximation, the induced PGB mass matrix is diagonal, with equal diagonal entries, 4 (0fi~t,~0)~ 2NN A2 ~ J D TC ETC

By comparison to

- 2NDNTC

(OI~AI0)2

(D.2)

Then, a~would be enhanced [31] in the expected fashion if (0I~//A~/’AI0) were, as would typically the expansion coefficients b~,c~,...,g 0°0°which enter into eq. (10.1).

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[371E.